Eoss — Verh-Fuucfious. 



41 



The rule di:ffers from that of algebraic division only in the fact 

 that each term of the quotient operates on the whole divisor instead of 

 being muUiplied into it. To find the first term of the quotient, we 

 ask what power of p operating on the first term of the divisor will 

 produce the first term of the dividend The answer is evidently ySl 

 Operating with this on the whole divisor (that is, squaring it) we 

 obtain the first subtrahend. Subtracting this we have the second 

 dividend. Again, operating with 3yS on the whole divisor we obtain 

 the second subtrahend, which gives the second remainder 4. IS'ow 

 since a quantity operating on a subject produces only itself (§ 5), the 

 number 4 in the quotient operating on the divisor will produce 

 nothing but itself, namely, the number 4 required to complete the 

 division. And the result may be verified by reversing the process and 

 operating with the quotient on the divisor, when the dividend will be 

 obtained. 



Or, we may arrest the division after the first term of the quotient 

 has been obtained, and then write the latter with a remainder, so that 

 it becomes 



3/3^+6^ + 19 

 ^ 13' + 2^ + 5 



The above is an example of division in descending terms of (3 ; but, 

 by reversing both divisor and dividend, we may obtain the quotient by 

 ascending division, thus : — 



5-^2fS + ft'J 44 + 26/3 +17/32+ 4/33+/3^ ^44 + 26 (^'^^ j + 4 J 



26^+17^^ + 4/3^ + 13' 

 26(3 +13P' 



4/3' + 4f3' + l3' 

 4^^ + 4/3' + (3' 



Here the first term of the quotient is 44, which merely reproduces 

 itself for the first subtrahend. For the second term of the quotient 



we have 26 — , since ^ ^ operating on the divisor reduces it to 



/3 + ^/3-, a form convenient for the process ; and for the same reason 

 the same operation appears in every term of the quotient, which may 

 therefore be written in the form 



[44 + 26^ + 4/32]^. 



